$A$ body of density $\rho$ is dropped from rest at a height $h$ into a lake of density $\delta$ $(\delta > \rho)$. Neglecting all dissipative forces,find the maximum depth to which the body sinks before returning to float on the surface.

$A$ body floats with $\frac{1}{n}$ of its volume outside of water. If the body is pushed to a depth $h$ inside the water and released, it will come to the surface after time $t$. Then:

Four identical beakers contain the same amount of water as shown below. Beaker $A$ contains only water. $A$ lead ball is held submerged in beaker $B$ by a string from above. $A$ same-sized plastic ball,say a table tennis $(TT)$ ball,is held submerged in beaker $C$ by a string attached to a stand from outside. Beaker $D$ contains a same-sized $TT$ ball which is held submerged by a string attached to the bottom of the beaker. These beakers (without the stand) are placed on weighing pans and register readings $w_{A}, w_{B}, w_{C}$,and $w_{D}$ for $A, B, C$,and $D$,respectively. Effects of the mass and volume of the stand and string are to be neglected.

$A$ metal cube is embedded inside a block of ice that is floating on the surface of water. The ice melts completely, and the metal cube sinks into the water. What happens to the water level in the container?

$A$ block of volume $V$ and density $\sigma_b$ is placed in a liquid of density $\sigma_l$ (where $\sigma_l > \sigma_b$). If the block is moved upward by a height $h$ while remaining completely submerged in the liquid,what is the increase in the gravitational potential energy of the system?

$A$ uniform rod of density $\rho$ is placed in a wide tank containing a liquid of density $\rho_0$ $(\rho_0 > \rho)$. The depth of the liquid in the tank is half the length of the rod. The rod is in equilibrium,with its lower end resting on the bottom of the tank. In this position,the rod makes an angle $\theta$ with the horizontal.